Two prime-related conjectures

[sweety439]

Originally Posted by sweety439

Conjecture 1 (conjecture about square numbers and odd primes): Every number which is not twice a square number (A001105) can be written as (twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 47 known counterexamples, the largest known counterexample is 43358, and I conjectured that all other numbers which is not twice a square number can be written as this form.

1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358

Conjecture 2 (conjecture about triangular numbers and odd primes): Every number which is not twice a triangular number (A002378) can be written as (twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 8 known counterexamples, the largest known counterexample is 432, and I conjectured that all other numbers which is not twice a triangular number can be written as this form.

1, 3, 4, 10, 14, 122, 422, 432

Now OEIS has sequences about these two conjectures!!! A347567 A347568

[sweety439]

Related research: perfect power (0 and 1 are not counted as perfect power) + prime (the even prime 2 is allowed)

OEIS sequences for this: A119748 A277075 A196228 A253238 A276711

In the past, I conjectured that all integers >24 can be written as perfect power (0 and 1 are not counted as perfect power) + prime, I tested dozens of numbers, and I found many numbers having only one way to write:

6 = 4 + 2
7 = 4 + 3
9 = 4 + 5
10 = 8 + 2
12 = 9 + 3
13 = 8 + 5
14 = 9 + 5
16 = 9 + 7
17 = 4 + 13
18 = 16 + 2
20 = 9 + 11
22 = 9 + 13
25 = 8 + 17
26 = 9 + 17
31 = 8 + 23
36 = 25 + 11
42 = 25 + 17
48 = 25 + 23
58 = 27 + 31
60 = 49 + 11
64 = 27 + 37
74 = 27 + 47
76 = 9 + 67
82 = 9 + 73
85 = 32 + 53
90 = 49 + 41
114 = 25 + 89
120 = 49 + 71
127 = 125 + 2
170 = 81 + 89
193 = 36 + 157
196 = 125 + 71
202 = 9 + 193
214 = 125 + 89
324 = 125 + 199
328 = 225 + 103
331 = 324 + 7
370 = 243 + 127
412 = 81 + 331
505 = 324 + 181
562 = 225 + 337
676 = 243 + 433
706 = 243 + 463
730 = 243 + 487
799 = 576 + 223
841 = 32 + 809
1024 = 27 + 997
1087 = 36 + 1051
1204 = 81 + 1123
1243 = 324 + 919
1681 = 128 + 1553
1849 = 128 + 1721
2146 = 9 + 2137
2293 = 1296 + 997
2986 = 125 + 2861
3319 = 128 + 3191
10404 = 343 + 10061
46656 = 46225 + 431
52900 = 35937 + 16963
112896 = 125 + 112771
122500 = 1331 + 121169

(I checked all numbers <= 2^24)

but I don't know why my conjecture fails at the number 1549, also 1771561 is another counterexample, it is known (checked by others), my conjecture works at all numbers <= 10^10 except 1549 and 1771561 (the small numbers cannot be written as this way is 1, 2, 3, 4, 5, 8, 24, thus the set of all numbers which cannot be written as this way is (likely) {1, 2, 3, 4, 5, 8, 24, 1549, 1771561}

(if only odd primes are allowed, and the even prime 2 is not allowed, then the set of all numbers is (likely) {1, 2, 3, 4, 5, 6, 8, 10, 18, 24, 127, 1549, 1771561}, the number 127 is interesting as it is the first odd number >3 which is not the sum of power of 2 and a prime)

[sweety439]

Compare with the numbers with only one way to write as:

"twice a positive square number" + "odd prime or twice an odd prime"

or

"twice a positive triangular number" + "odd prime or twice an odd prime"

For the former (square number):

5 = 1*3 + 2*1^2
7 = 1*5 + 2*1^2
9 = 1*7 + 2*1^2
11 = 1*3 + 2*2^2
12 = 2*5 + 2*1^2
14 = 2*3 + 2*2^2
16 = 2*7 + 2*1^2
22 = 2*7 + 2*2^2
23 = 1*5 + 2*3^2
27 = 1*19 + 2*2^2
29 = 1*11 + 2*3^2
30 = 2*11 + 2*2^2
33 = 1*31 + 2*1^2
34 = 2*13 + 2*2^2
36 = 2*17 + 2*1^2
38 = 2*3 + 2*4^2
41 = 1*23 + 2*3^2
44 = 2*13 + 2*3^2
47 = 1*29 + 2*3^2
48 = 2*23 + 2*1^2
52 = 2*17 + 2*3^2
53 = 1*3 + 2*5^2
57 = 1*7 + 2*5^2
58 = 2*13 + 2*4^2
59 = 1*41 + 2*3^2
65 = 1*47 + 2*3^2
71 = 1*53 + 2*3^2
80 = 2*31 + 2*3^2
83 = 1*11 + 2*6^2
86 = 2*7 + 2*6^2
92 = 2*37 + 2*3^2
95 = 1*23 + 2*6^2
100 = 2*41 + 2*3^2
102 = 2*47 + 2*2^2
107 = 1*89 + 2*3^2
110 = 2*19 + 2*6^2
113 = 1*41 + 2*6^2
123 = 1*73 + 2*5^2
140 = 2*61 + 2*3^2
143 = 1*71 + 2*6^2
146 = 2*37 + 2*6^2
148 = 2*73 + 2*1^2
149 = 1*131 + 2*3^2
152 = 2*67 + 2*3^2
158 = 2*43 + 2*6^2
161 = 1*89 + 2*6^2
164 = 2*73 + 2*3^2
188 = 2*13 + 2*9^2
194 = 2*61 + 2*6^2
197 = 1*179 + 2*3^2
198 = 2*83 + 2*4^2
212 = 2*97 + 2*3^2
218 = 2*73 + 2*6^2
230 = 2*79 + 2*6^2
233 = 1*71 + 2*9^2
239 = 1*167 + 2*6^2
240 = 2*71 + 2*7^2
257 = 1*239 + 2*3^2
266 = 2*97 + 2*6^2
272 = 2*127 + 2*3^2
278 = 2*103 + 2*6^2
281 = 1*263 + 2*3^2
284 = 2*61 + 2*9^2
287 = 1*269 + 2*3^2
290 = 2*109 + 2*6^2
302 = 2*7 + 2*12^2
308 = 2*73 + 2*9^2
314 = 2*13 + 2*12^2
317 = 1*29 + 2*12^2
318 = 2*59 + 2*10^2
323 = 1*251 + 2*6^2
332 = 2*157 + 2*3^2
340 = 2*89 + 2*9^2
347 = 1*59 + 2*12^2
356 = 2*97 + 2*9^2
362 = 2*37 + 2*12^2
368 = 2*103 + 2*9^2
383 = 1*311 + 2*6^2
386 = 2*157 + 2*6^2
404 = 2*193 + 2*3^2
407 = 1*389 + 2*3^2
410 = 2*61 + 2*12^2
413 = 1*251 + 2*9^2
422 = 2*67 + 2*12^2
438 = 2*23 + 2*14^2
442 = 2*157 + 2*8^2
443 = 1*281 + 2*9^2
446 = 2*79 + 2*12^2
449 = 1*431 + 2*3^2
458 = 2*193 + 2*6^2
470 = 2*199 + 2*6^2
482 = 2*97 + 2*12^2
492 = 2*197 + 2*7^2
500 = 2*241 + 2*3^2
506 = 2*109 + 2*12^2
530 = 2*229 + 2*6^2
536 = 2*43 + 2*15^2
542 = 2*127 + 2*12^2
548 = 2*193 + 2*9^2
554 = 2*241 + 2*6^2
566 = 2*139 + 2*12^2
569 = 1*281 + 2*12^2
590 = 2*151 + 2*12^2
596 = 2*73 + 2*15^2
602 = 2*157 + 2*12^2
620 = 2*229 + 2*9^2
626 = 2*277 + 2*6^2
632 = 2*307 + 2*3^2
638 = 2*283 + 2*6^2
650 = 2*181 + 2*12^2
656 = 2*103 + 2*15^2
662 = 2*7 + 2*18^2
668 = 2*109 + 2*15^2
680 = 2*331 + 2*3^2
692 = 2*337 + 2*3^2
698 = 2*313 + 2*6^2
743 = 1*293 + 2*15^2
773 = 1*701 + 2*6^2
782 = 2*67 + 2*18^2
785 = 1*137 + 2*18^2
788 = 2*313 + 2*9^2
794 = 2*73 + 2*18^2
798 = 2*383 + 2*4^2
818 = 2*373 + 2*6^2
824 = 2*331 + 2*9^2
848 = 2*199 + 2*15^2
863 = 1*701 + 2*9^2
872 = 2*211 + 2*15^2
884 = 2*433 + 2*3^2
890 = 2*409 + 2*6^2
926 = 2*139 + 2*18^2
938 = 2*433 + 2*6^2
980 = 2*409 + 2*9^2
998 = 2*463 + 2*6^2
1010 = 2*181 + 2*18^2
1022 = 2*367 + 2*12^2
1082 = 2*397 + 2*12^2
1094 = 2*223 + 2*18^2
1118 = 2*523 + 2*6^2
1124 = 2*337 + 2*15^2
1148 = 2*349 + 2*15^2
1172 = 2*577 + 2*3^2
1178 = 2*13 + 2*24^2
1220 = 2*601 + 2*3^2
1227 = 1*1129 + 2*7^2
1232 = 2*607 + 2*3^2
1238 = 2*43 + 2*24^2
1292 = 2*421 + 2*15^2
1322 = 2*337 + 2*18^2
1367 = 1*719 + 2*18^2
1388 = 2*613 + 2*9^2
1415 = 1*263 + 2*24^2
1418 = 2*673 + 2*6^2
1478 = 2*163 + 2*24^2
1502 = 2*607 + 2*12^2
1562 = 2*457 + 2*18^2
1586 = 2*757 + 2*6^2
1598 = 2*223 + 2*24^2
1622 = 2*487 + 2*18^2
1668 = 2*809 + 2*5^2
1670 = 2*691 + 2*12^2
1703 = 1*821 + 2*21^2
1748 = 2*433 + 2*21^2
1754 = 2*733 + 2*12^2
1787 = 1*1499 + 2*12^2
1828 = 2*73 + 2*29^2
1844 = 2*193 + 2*27^2
1892 = 2*937 + 2*3^2
1898 = 2*373 + 2*24^2
1940 = 2*241 + 2*27^2
1958 = 2*79 + 2*30^2
1988 = 2*769 + 2*15^2
2042 = 2*877 + 2*12^2
2060 = 2*1021 + 2*3^2
2090 = 2*1009 + 2*6^2
2123 = 1*971 + 2*24^2
2132 = 2*337 + 2*27^2
2138 = 2*1033 + 2*6^2
2174 = 2*1051 + 2*6^2
2180 = 2*1009 + 2*9^2
2210 = 2*1069 + 2*6^2
2234 = 2*541 + 2*24^2
2328 = 2*1163 + 2*1^2
2342 = 2*271 + 2*30^2
2402 = 2*877 + 2*18^2
2408 = 2*1123 + 2*9^2
2438 = 2*643 + 2*24^2
2486 = 2*919 + 2*18^2
2507 = 1*1049 + 2*27^2
2558 = 2*379 + 2*30^2
2582 = 2*967 + 2*18^2
2648 = 2*883 + 2*21^2
2708 = 2*1129 + 2*15^2
2732 = 2*277 + 2*33^2
2762 = 2*1237 + 2*12^2
2768 = 2*1303 + 2*9^2
2822 = 2*1087 + 2*18^2
2858 = 2*853 + 2*24^2
2900 = 2*1009 + 2*21^2
2933 = 1*2861 + 2*6^2
3002 = 2*601 + 2*30^2
3062 = 2*631 + 2*30^2
3110 = 2*1231 + 2*18^2
3242 = 2*1297 + 2*18^2
3284 = 2*1201 + 2*21^2
3317 = 1*3299 + 2*3^2
3434 = 2*421 + 2*36^2
3452 = 2*997 + 2*27^2
3482 = 2*1597 + 2*12^2
3515 = 1*2633 + 2*21^2
3530 = 2*1621 + 2*12^2
3572 = 2*1777 + 2*3^2
3620 = 2*1801 + 2*3^2
3662 = 2*67 + 2*42^2
3713 = 1*1913 + 2*30^2
3722 = 2*97 + 2*42^2
3758 = 2*1303 + 2*24^2
3770 = 2*1741 + 2*12^2
3962 = 2*1657 + 2*18^2
3980 = 2*1549 + 2*21^2
3998 = 2*1423 + 2*24^2
4022 = 2*1867 + 2*12^2
4082 = 2*277 + 2*42^2
4118 = 2*1483 + 2*24^2
4148 = 2*1993 + 2*9^2
4178 = 2*2053 + 2*6^2
4292 = 2*2137 + 2*3^2
4334 = 2*2131 + 2*6^2
4490 = 2*1669 + 2*24^2
4502 = 2*487 + 2*42^2
4532 = 2*241 + 2*45^2
4538 = 2*1693 + 2*24^2
4568 = 2*2203 + 2*9^2
4586 = 2*997 + 2*36^2
4673 = 1*2081 + 2*36^2
4688 = 2*823 + 2*39^2
4820 = 2*1321 + 2*33^2
4832 = 2*1327 + 2*33^2
4958 = 2*1579 + 2*30^2
5078 = 2*2503 + 2*6^2
5102 = 2*787 + 2*42^2
5300 = 2*1129 + 2*39^2
5612 = 2*2797 + 2*3^2
5642 = 2*2677 + 2*12^2
5708 = 2*829 + 2*45^2
5798 = 2*1999 + 2*30^2
5852 = 2*2917 + 2*3^2
5942 = 2*2647 + 2*18^2
5987 = 1*2459 + 2*42^2
6008 = 2*1483 + 2*39^2
6188 = 2*1069 + 2*45^2
6218 = 2*193 + 2*54^2
6302 = 2*2251 + 2*30^2
6332 = 2*2437 + 2*27^2
6368 = 2*1663 + 2*39^2
6518 = 2*2683 + 2*24^2
6602 = 2*997 + 2*48^2
6797 = 1*6779 + 2*3^2
6836 = 2*2689 + 2*27^2
6938 = 2*3433 + 2*6^2
7004 = 2*3061 + 2*21^2
7142 = 2*2671 + 2*30^2
7622 = 2*211 + 2*60^2
7718 = 2*3823 + 2*6^2
7730 = 2*3541 + 2*18^2
7928 = 2*3739 + 2*15^2
7982 = 2*3847 + 2*12^2
8432 = 2*967 + 2*57^2
8444 = 2*1621 + 2*51^2
8558 = 2*4243 + 2*6^2
8660 = 2*3889 + 2*21^2
8828 = 2*2389 + 2*45^2
9008 = 2*4423 + 2*9^2
9020 = 2*541 + 2*63^2
9122 = 2*2797 + 2*42^2
9290 = 2*2341 + 2*48^2
9308 = 2*2053 + 2*51^2
9422 = 2*4567 + 2*12^2
9722 = 2*2557 + 2*48^2
9860 = 2*4201 + 2*27^2
10964 = 2*3457 + 2*45^2
11012 = 2*5281 + 2*15^2
11090 = 2*4969 + 2*24^2
11498 = 2*2833 + 2*54^2
11972 = 2*2017 + 2*63^2
12062 = 2*3727 + 2*48^2
12098 = 2*1693 + 2*66^2
12548 = 2*3673 + 2*51^2
12602 = 2*1117 + 2*72^2
12878 = 2*2083 + 2*66^2
14018 = 2*4093 + 2*54^2
14162 = 2*997 + 2*78^2
14882 = 2*7297 + 2*12^2
15758 = 2*823 + 2*84^2
15908 = 2*7873 + 2*9^2
16172 = 2*6997 + 2*33^2
16838 = 2*5503 + 2*54^2
17168 = 2*3823 + 2*69^2
17648 = 2*8599 + 2*15^2
18428 = 2*9133 + 2*9^2
19142 = 2*1471 + 2*90^2
20330 = 2*3109 + 2*84^2
20918 = 2*9883 + 2*24^2
21548 = 2*10333 + 2*21^2
21722 = 2*457 + 2*102^2
23018 = 2*2293 + 2*96^2
23612 = 2*9781 + 2*45^2
25022 = 2*6427 + 2*78^2
27668 = 2*8209 + 2*75^2
30212 = 2*7537 + 2*87^2
30668 = 2*13309 + 2*45^2
31130 = 2*15241 + 2*18^2
32162 = 2*15937 + 2*12^2
32372 = 2*7537 + 2*93^2
47702 = 2*6427 + 2*132^2
63758 = 2*28279 + 2*60^2
66410 = 2*6961 + 2*162^2
89072 = 2*11047 + 2*183^2

For the latter (triangular number):

5 = 1*3 + 1*2
7 = 1*5 + 1*2
8 = 2*3 + 1*2
11 = 1*5 + 2*3
18 = 2*3 + 3*4
21 = 1*19 + 1*2
22 = 2*5 + 3*4
24 = 2*11 + 1*2
27 = 1*7 + 4*5
32 = 2*13 + 2*3
38 = 2*13 + 3*4
50 = 2*19 + 3*4
51 = 1*31 + 4*5
54 = 2*17 + 4*5
57 = 1*37 + 4*5
60 = 2*29 + 1*2
62 = 2*3 + 7*8
74 = 2*31 + 3*4
84 = 2*41 + 1*2
105 = 1*103 + 1*2
108 = 2*53 + 1*2
111 = 1*109 + 1*2
126 = 2*53 + 4*5
140 = 2*67 + 2*3
150 = 2*47 + 7*8
174 = 2*59 + 7*8
180 = 2*89 + 1*2
186 = 2*83 + 4*5
242 = 2*43 + 12*13
252 = 2*71 + 10*11
258 = 2*101 + 7*8
270 = 2*107 + 7*8
357 = 1*337 + 4*5
372 = 2*131 + 10*11
471 = 1*199 + 16*17
492 = 2*191 + 10*11
510 = 2*227 + 7*8
630 = 2*179 + 16*17
666 = 2*197 + 16*17
690 = 2*317 + 7*8
765 = 1*709 + 7*8
792 = 2*71 + 25*26
810 = 2*269 + 16*17
1080 = 2*449 + 13*14
1112 = 2*541 + 5*6
1380 = 2*599 + 13*14
1434 = 2*311 + 28*29
1602 = 2*773 + 7*8
1848 = 2*599 + 25*26
1920 = 2*257 + 37*38
2160 = 2*827 + 22*23
3726 = 2*1367 + 31*32
4752 = 2*1151 + 49*50
5397 = 1*1237 + 64*65
5652 = 2*1601 + 49*50
7800 = 2*2819 + 46*47
12420 = 2*5507 + 37*38
16632 = 2*3851 + 94*95

Like "perfect power (not including 0 and 1) + prime", it is conjectured that 89072 and 16632 are the largest examples of them, respectively.

[sweety439]

There are complement of A065377 and A065397 of the primes A000040 in OEIS: A065376 and A065396, respectively.

[sweety439]

The conjecture 2 in post #1 (i.e. A347568 is full with 8 terms: {1, 3, 4, 10, 14, 122, 422, 432}):

Zhi-Wei Sun, Conjectures on sums of primes and triangular numbers

(page 3 has "furthermore, any positive integer n (not∈) {2, 5, 7, 61, 211, 216} can be written in the form p + Tx with x ∈ Z+, where p is an odd prime or zero", which is equivalent to the only even numbers in A347568 are {4, 10, 14, 122, 422, 432}) and (page 5 has "Any odd integer n > 3 can be written in the form p + x(x + 1) with p a prime and and x a positive integer", which is equivalent to the only odd numbers in A347568 are {1, 3})

The conjecture 1 in post #1 (i.e. A347567 is full with 47 terms: {1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}):

G. H. Hardy, J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes (see the attached pdf file)

(page 49 has "Conjecture H. Every large number n is either a square or the sum of a prime and a square.", which is called "Hardy & Littlewood's Conjecture H" (not to be confused with Hardy–Littlewood conjecture about prime k-tuples), and is equivalent to A347567 contains only finitely many even numbers (or finitely many even numbers which are twice composite numbers, if 0 is counted as square in that article), also "probably because, the idea that every number is a square, or the sum of a prime and a square, is refuted (even if I is counted as a prime) by such immediate examples as 34 and 58. But the problem of the representation of an odd number in the form t+2m^2 has received some attention; and it has been verified that the only odd numbers less than 9000, and not of the form desired, are 5777 and 5993", which is related to the even numbers in A347567 and the odd numbers in A347567, respectively)

[sweety439]

For A347567, 2*square+p, where p is either odd prime or twice an odd prime, the set is:

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}

If p = 4 (twice the even prime 2) is allowed, then the set become:

{1, 3, 4, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358} (the same set except the number 6)

If p = 1, 2, 4 are all allowed, then the set become:

{1, 17, 26, 62, 68, 116, 122, 137, 170, 182, 227, 254, 260, 428, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}

If square = 0 is allowed, then the set become:

{1, 4, 20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5777, 5972, 5993, 7352, 19268, 43358}

If square = 0 and p = 4 (twice the even prime 2) are both allowed, then the set become:

{1, 4, 20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5777, 5972, 5993, 7352, 19268, 43358} (the same set as the previous)

If square = 0 and p = 1, 2, 4 are all allowed, then the set become:

{68, 116, 170, 182, 260, 428, 740, 1052, 1412, 1542, 2510, 2702, 2828, 3812, 5777, 5972, 5993, 7352, 19268, 43358}

[sweety439]

The numbers in A347567:

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}

Categories:

odd unit: {1}
odd primes: {3, 17, 137, 227, 977, 1187, 1493} (A042978 Stern primes, except the "oddest" prime 2)
odd composites: {5777, 5993} (counterexamples of the less-known Goldbach conjecture, composites in A060003)
even number whose half is unit: {} (the only such number is 2, but 2 is twice a square thus not considered)
even numbers whose half are primes: {4, 10, 26, 62, 74, 122, 254, 758, 878, 1142, 1658, 1982, 3098, 6638, 15098} (2*A065377)
even numbers whose half are composites: {20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5972, 7352, 19268, 43358} (2*A020495)

Factorizations:

1 = unit
3 = 3
4 = 2^2
6 = 2 * 3
10 = 2 * 5
17 = 17
20 = 2^2 * 5
26 = 2 * 13
62 = 2 * 31
68 = 2^2 * 17
74 = 2 * 37
116 = 2^2 * 29
122 = 2 * 61
137 = 137
170 = 2 * 5 * 17
182 = 2 * 7 * 13
227 = 227
254 = 2 * 127
260 = 2^2 * 5 * 13
428 = 2^2 * 107
452 = 2^2 * 113
740 = 2^2 * 5 * 37
758 = 2 * 379
878 = 2 * 439
977 = 977
1052 = 2^2 * 263
1142 = 2 * 571
1187 = 1187
1412 = 2^2 * 353
1460 = 2^2 * 5 * 73
1493 = 1493
1542 = 2 * 3 * 257
1658 = 2 * 829
1982 = 2 * 991
2510 = 2 * 5 * 251
2702 = 2 * 7 * 193
2828 = 2^2 * 7 * 101
3098 = 2 * 1549
3812 = 2^2 * 953
5777 = 53 * 109
5972 = 2^2 * 1493
5993 = 13 * 461
6638 = 2 * 3319
7352 = 2^3 * 919
15098 = 2 * 7549
19268 = 2^2 * 4817
43358 = 2 * 7 * 19 * 163

Modulos:

Mod 2: {1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0} (nothing interesting, the only interesting thing is that "0" is much more than "1")

Mod 3: {1, 0, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2} (all are "2" after 10, with only an exception of 1542, which is "0", see the reference in Prime Curios!)

Mod 4: {1, 3, 0, 2, 2, 1, 0, 2, 2, 0, 2, 0, 2, 1, 2, 2, 3, 2, 0, 0, 0, 0, 2, 2, 1, 0, 2, 3, 0, 0, 1, 2, 2, 2, 2, 2, 0, 2, 0, 1, 0, 1, 2, 0, 2, 0, 2} (the same as mod 2, most are "0" or "2", and "0" is more than "2" (most "2" occurs for twice primes, see A308516), "1" is more than "3" (the only "3" are 3, 227, 1187))

Mod 5: {1, 3, 4, 1, 0, 2, 0, 1, 2, 3, 4, 1, 2, 2, 0, 2, 2, 4, 0, 3, 2, 0, 3, 3, 2, 2, 2, 2, 2, 0, 3, 2, 3, 2, 0, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 3, 3} (all are "2" or "3" after 2510, and all are "0", "2", or "3" after 254)

Mod 6: {1, 3, 4, 0, 4, 5, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 5, 2, 2, 5, 0, 2, 2, 2, 2, 2, 2, 2, 5, 2, 5, 2, 2, 2, 2, 2} (most are "2", and the interesting thing the same as mod 3, all are "2" or "5" after 10, with only an exception of 1542, which is "0", the only "3" is for the number 3, see the Graham comment in A060003)

Mod 7: {1, 3, 4, 6, 3, 3, 6, 5, 6, 5, 4, 4, 3, 4, 2, 0, 3, 2, 1, 1, 4, 5, 2, 3, 4, 2, 1, 4, 5, 4, 2, 2, 6, 1, 4, 0, 0, 4, 4, 2, 1, 1, 2, 2, 6, 4, 0} (nothing interesting)

Mod 8: {1, 3, 4, 6, 2, 1, 4, 2, 6, 4, 2, 4, 2, 1, 2, 6, 3, 6, 4, 4, 4, 4, 6, 6, 1, 4, 6, 3, 4, 4, 5, 6, 2, 6, 6, 6, 4, 2, 4, 1, 4, 1, 6, 0, 2, 4, 6} (most are "2", "4", "6", there is only one "0", also there is no "7", see A308516 and A317966)

Mod 9: {1, 3, 4, 6, 1, 8, 2, 8, 8, 5, 2, 8, 5, 2, 8, 2, 2, 2, 8, 5, 2, 2, 2, 5, 5, 8, 8, 8, 8, 2, 8, 3, 2, 2, 8, 2, 2, 2, 5, 8, 5, 8, 5, 8, 5, 8, 5} (the same as mod 3, most are "2", "5", "8", 9 is the smallest mod such that there is no "0", the next such mod is 11)

Mod 10: {1, 3, 4, 6, 0, 7, 0, 6, 2, 8, 4, 6, 2, 7, 0, 2, 7, 4, 0, 8, 2, 0, 8, 8, 7, 2, 2, 7, 2, 0, 3, 2, 8, 2, 0, 2, 8, 8, 2, 7, 2, 3, 8, 2, 8, 8, 8} (the same as mod 5, all are "2", "3", "7", "8" after 2510, and all are "0", "2", "3", "7", "8" after 254, there is no "5")

Mod 11: {1, 3, 4, 6, 10, 6, 9, 4, 7, 2, 8, 6, 1, 5, 5, 6, 7, 1, 7, 10, 1, 3, 10, 9, 9, 7, 9, 10, 4, 8, 8, 2, 8, 2, 2, 7, 1, 7, 6, 2, 10, 9, 5, 4, 6, 7, 7} (nothing interesting)

Mod 12: {1, 3, 4, 6, 10, 5, 8, 2, 2, 8, 2, 8, 2, 5, 2, 2, 11, 2, 8, 8, 8, 8, 2, 2, 5, 8, 2, 11, 8, 8, 5, 6, 2, 2, 2, 2, 8, 2, 8, 5, 8, 5, 2, 8, 2, 8, 2} (the same as mod 4 and mod 6, all are "2", "5", "8", "11" after 10, with only an exception of 1542, which is "6")

(In general, possible residues of 2*m^2 mod n are more than impossible residues of 2*m^2 mod n, to the list of mod n)

[sweety439]

Related conjectures:

11 is the only non-generalized pentagonal number which cannot be written as sum of a prime and a positive generalized pentagonal number.

besides 1 and 3, 79 is the only odd number which cannot be written as sum of a prime and twice a positive generalized pentagonal number.

and the only non-generalized pentagonal numbers can be written as sum of a prime and a positive generalized pentagonal number in only one way are {3, 6, 13, 16, 21, 23, 27, 47, 50, 61, 67, 127, 211}

[sweety439]

Conjecture 1: Every natural number n > 21679 is a square or the sum of a square and a prime number.

Conjecture 2: Every prime number p > 7549 is the sum of a nonzero square and a prime number.

Conjecture 3: Every odd natural number n > 5993 is the sum of twice a square and a prime number.

Conjecture 4: Every odd prime number p > 1493 is the sum of twice a nonzero square and a prime number.

Conjecture 5: Every natural number n > 216 is a triangular number or the sum of a triangular number and a prime number.

Conjecture 6: Every prime number p > 211 is the sum of a nonzero triangular number and a prime number.

Conjecture 7: Every odd natural number n > 1 is the sum of twice a triangular number and a prime number.

Conjecture 8: Every odd prime number p > 3 is the sum of twice a nonzero triangular number and a prime number.